large time behavior of periodic viscosity solutions of...

Large Time Behavior of Periodic Viscosity Solutions ofIntegro-differential Equations

Adina CIOMAGAjoint work with Guy Barles, Emmanuel Chasseigne and Cyril Imbert

Universite Paris Diderot,Laboratoire Jacques Louis Lions

June 1, 2016

Adina Ciomaga (Universite Paris Diderot) HJnet Final Conference, Rennes June 1, 2016 1 / 27

Outline

1 Framework: Integro-Differential Equations

2 Large Time Behavior

3 Strong Maximum Principle

4 Holder and Lipschitz Regularity


Framework

Partial Integro Differential Equations (PIDEs)

Problem

Long Time Behavior of solutions of Partial Integro Differential Equations (PIDEs)ut + F (x ,Du,D2u, I[x , u]) = 0, in Rd × (0,+∞)

u(x , 0) = u0(x), in Rd(1)

The nonlinearity F is degenerate elliptic, i.e.

F (x , p,X , l1) ≤ F (x , p,Y , l2) if X ≥ Y , l1 ≥ l2, (E)

I[x , u] is an integro-differential operator of the form

I[x , u] =

∫Rd

(u(x + z , t)− u(x , t)− Du(x , t) · z1B (z))µx (dz)

(µx )x family of Levy measures s.t. supx

∫Rd min(1, |z |2)µx (dz) <∞.


Framework


Problem


u(x , 0) = u0(x), in Rd(1)



I[x , u] is a Levy-Ito operator of the form

J [x , u] =

∫Rd

(u(x + j(x , z))− u(x)− Du(x) · j(x , z)1B (z))µ(dz)

with µ a Levy measure and j(x , z) the size of the jumps at x .


Framework


Problem


u(x , 0) = u0(x), in Rd(1)



Fractional Laplacian of order β ∈ (0, 2)

(−∆)β/2u =

∫Rd

(u(x + z , t)− u(x , t)− Du(x , t) · z1B (z))dz

|z |d+β


Framework

Partial Integro-Differential Equations. Levy Processes

Figure: Stable Levy process. Jump discontinuities are represented by vertical lines.

The infinitezimal generator of a Levy process

Lu(x) = b · Du︸︷︷︸drift

+ tr(AD2u)︸︷︷︸diffusion

+

∫Rd

(u(x + j(x , z))− u(x)− Du · j(x , z)1B (z)

)µ(dz)︸︷︷︸

jumps

.


Framework


Figure: Stable Levy process. Jump discontinuities are represented by vertical lines.

The infinitezimal generator of a wider class of Markov processes of Courrege form

Lu(x) = c(x)u(x)︸︷︷︸killing

+ b(x) · Du︸︷︷︸drift

+ tr(A(x)D2u)︸︷︷︸diffusion

+

∫Rd

(u(x + z)− u(x)− Du · z K (x , z)

)µx (dz)︸︷︷︸

jumps

.


Framework


Example (Drift diffusion PIDEs)

ut + (−∆)β/2u + b(x) · Du = f (x)

Example (Mixed PIDEs)

ut −∆x1u + (−∆x2 )β/2u + b(x) · Du = f (x)

with 1 < β < 2.

Example (Coercive PIDEs)

ut−tr(A(x)D2u)− I[u] + b(x)|Du|m = f (x)

whereA(x) ≥ a(x)I ≥ 0, b(x) ≥ b0 > 0,m > 2.


LTB

LargeTime Behavior - periodic setting

Problem

Establish the long time behavior of periodic viscosity solutions:

u(x , t) = λt + v(x) + ot(1), as t →∞.

Nonlocal: Imbert, Monneau, Rouy ’07

Parabolic PDEs: Barles, Mitake, Ishii ’09, Barles and Souganidis ’06, ’01,Barles Da Lio ’05, Dirr and Souganidis, ’95, Roquejoffre ’01

Hamilton Jacobi equations: Barles, Roquejoffre ’06, Barles Souganidis ’00Lions, Papanicolau, Varadhan, Ishii’ 10, Namah and Roquejoffre ’99, Fathi’98, Fathi and Mather ’00, Arisawa ’97


LTB

LargeTime Behavior - periodic setting

Theorem (Barles, Chasseigne, C., Imbert ’13)

Under suitable assumptions, the solution of the initial value problem

ut + F (x ,D2u, I[u]) + H(x ,Du) = f (x). (2)

with u0 ∈ C 0,α and Zd periodic satisfies

u(x , t)− λt → v(x), as t∞ uniformly in x,

where v is the unique periodic solution (up to addition of constants) of thestationary ergodic problem

F (x ,D2v , I[v ]) + H(x ,Dv) = f (x)− λ in Rd . (3)

The study relies in general on two main ingredients:

Strong Maximum Principle

Regularity of viscosity solutions


LTB

The Ergodic Problem

To solve the ergodic problem

λ+ F (x ,D2v , I[v ]) + H(x ,Dv) = f (x) in Rd , (4)

use the classical approximation

δvδ + F (x ,D2vδ, I[vδ]) + H(x ,Dvδ) = f (x). (5)

Perron’s method and comparison principles: there exists a unique, bounded,periodic solution vδ s.t. ||δvδ||∞ ≤ C , hence δvδ(0)→ λ as δ → 0.

Lemma

vδ(x) = vδ(x)− vδ(0)

is uniformly bounded and equicontinuous, hence vδ → v (along subsequences).


LTB

The Ergodic Problem - Compactness Mixed PIDEs

Proof.

Argue by contradiction: assume ||vδ||∞ =: cδ →∞ as δ → 0. Then therenormalized functions wδ = vδ/cδ solve

δwδ +1

cδF (x , cδD

2wδ, cδI[wδ]) +1

cδH(x , cδDw

δ) =f (x)− δvδ(0)

cδ.

Since ||wδ|| = 1, solutions are uniformly C 0,α and up to a subsequence

wδ → w , as δ → 0, uniformly in x.

The limit w is Zd periodic, C 0,α, w(0) = 0, ||w ||∞ = 1 and satisfies

F (x ,D2w , I[w ]) + H(x ,Dw) = 0. (6)

Provided the limiting equation satisfies Strong Maximum Principle, w ≡ 0!


LTB

The Ergodic Problem - Compactness Mixed PIDEs

Proof.

Argue by contradiction: assume ||vδ||∞ =: cδ →∞ as δ → 0. Then therenormalized functions wδ = vδ/cδ solve

δwδ + F (x ,D2wδ, I[wδ]) +1

cδH(x , cδDw

δ) =f (x)− δvδ(0)

cδ.

Since ||wδ|| = 1, solutions are uniformly C 0,α and up to a subsequence



F (x ,D2w , I[w ]) + H(x ,Dw) = 0. (6)

Provided the limiting equation satisfies Strong Maximum Principle, w ≡ 0!


LTB

The Ergodic Problem - Compactness Coercive PIDEs

Proof.

To fix ideas, letH(x , p) = b(x)|p|m for m > 1.

The renormalized functions wδ = vδ/cδ solve

1

cm−1δ

δwδ +1

cm−1δ

F (x , cδD2wδ, cδI[wδ]) + b(x)|Dwδ|m =

f (x)− δvδ(0)

cmδ

.


b(x)|Dw |m = 0.

Provided b(x) ≥ b0 > 0 we get Dw ≡ 0, hence w ≡ 0!


LTB

The Ergodic Problem - Compactness Coercive PIDEs

Proof.

To fix ideas, letH(x , p) = b(x)|p|m for m > 1.

The renormalized functions wδ = vδ/cδ solve

1

cm−1δ

δwδ +1

cm−1δ

F (x , cδD2wδ, cδI[wδ]) + b(x)|Dwδ|m =

f (x)− δvδ(0)

cmδ

.

Solutions are uniformly C 0,α and up to a subsequence



b(x)|Dw |m = 0.

Provided b(x) ≥ b0 > 0 we get Dw ≡ 0, hence w ≡ 0!


LTB

The Convergence

Proof.

Both u(x , t) and v(x) + λt are solutions of (2). By comparison

m(t) := supx

(u(x , t)− λt − v(x)) m, as →∞.

Take then the Zd periodic functions w(x , t) = u(x , t)− λt and showw(x , t + tn)→ w(x , t) as tn →∞, where w solves

wt + F (x ,D2w , I[w ]) + H(x ,Dw) = f (x)− λ, in Rd × (0,+∞)

w(x , 0) = v(x), in Rd(7)

Passing to the limit in m(t + tn) as n→∞ m = supx (w(x , t)− v(x)) By theStrong Maximum Principle for the evolution equation above we get

w(x , t) = v(x) + m.

The conclusion follows.



Strong Maximum Principle for PIDEs

Problem

Establish Strong Maximum Principle for Dirichlet boundary value problemsut + F (x , t,Du,D2u,J [x , u]) = 0, in Ω× (0,T )u = ϕ on Ωc × [0,T ].

(8)

Strong Maximum Principle and (Strong) Comparison Results

nonlocal operators: Coville ’08;

elliptic second order equations:Bardi-Da Lio ’01, ’03, Da Lio ’04, Nirenberg ’53, Hopf ’20s;

* comparison results and Jensen-Ishii’s lemma:Jakobsen-Karlsen ’06, Barles-Imbert ’08, Jensen ’88, Ishii ’89, Crandall, Ishii,Lions ’90s.




Theorem (C. ’11)

Anu u ∈ USC (Rd × [0,T ]) viscosity subsolution of (8) that attains a maximumat (x0, t0) ∈ Ω× (0,T ) is constant in Ω× [0, t0].

Figure: SMaxP = horizontal and vertical propagation of maxima.



Horizontal Propagation - Translations of Measure Supports

Theorem (C. ’11)

If u attains a global maximum at (x0, t0) ∈ Rd × (0,T ), then u is constant on⋃n≥0 An × t0 with

A0 = x0, An+1 =⋃

x∈An

(x + supp(µx )). (9)

µ(dz) =dz

|z |d+β.



Horizontal Propagation - Translations of Measure Supports

Example (Measures supported in the unit ball)

µ(dz) = 1B (z)dz

|z |d+β.

Example (Measures charging two axis meeting at the origin)

µx (dz) = 1z1=±αz2(z)νx (dz),

Example (Pittfall of fractional diffusion on half space)

µ(dz) = 1z1≥0(z)dz

|z |d+β.



Horizontal Propagation - Nondegeneracy of the Measure

For any x ∈ Ω there exist β ∈ (1, 2), η ∈ (0, 1)and a constant Cµ(η) > 0 s.t. for 0 < |p| < R∫

Cη,γ(p)

|z |2µx (dz) ≥ Cµ(η)γβ−2,∀γ ≥ 1

Cη,γ(p) = z ; (1− η)|z ||p| ≤ |p · z | ≤ 1/γ

Theorem (C. ’11)

Under suitable nondegeneracy and scaling assumptions, if a usc viscositysubsolution u attains a maximum at P0 = (x0, t0), then u is constant in thehorizontal component of the domain, passing through point P0.




Example (Overcome fractional diffusion on halfspace)

µ(dz) = 1z1≥0(z)dz

|z |d+β, β > 1.

Theorem (C. ’11)

Under suitable nondegeneracy and scaling assumptions, if a usc viscositysubsolution u attains a maximum at P0 = (x0, t0), then u is constant in thehorizontal component of the domain, passing through point P0.




Example (SMaxP driven by differential terms)

ut − tr(σ(x)σ∗(x)D2u)−c(x)I[x , u] = f (x), in Ω× (0,T )

where σ is a positive definite matrix and c(x) ≥ 0.

Example (SMaxP driven by the nonlocal term)

ut + b(x) · Du + (−∆xu)β/2 = f (x), in Ω× (0,T )

where b(·) is a bounded vector field and the fractional exponent β > 1.

Example (SMaxP for mixed differential-nonlocal terms)

ut − a1(x)∆x1u + a2(x)(−∆x2u)β/2 = f (x), in Ω× (0,T )

where a1(x), a2(x) ≥ a0 > 0 and the fractional exponent β > 1.



Strong Comparison Principle

Theorem (C.)

If u and v are an usc viscosity subsolution, lsc viscosity supersolution s.t. u − vattains a maximum at P0, then u − v is constant in S(P0).

Example (Mixed PIDEs)

ut − a1(x)∆x1u + a2(x)(−∆x2 )β/2u = f (x)

if the fractional exponent β > 1 and

a1(x), a2(x) ≥ a0 > 0.

Example (Coercive PIDEs)

ut − I[u] + b(x)|Du|m = f (x)

if u is Lipschitz continuous, b(x) ≥ b0 > 0 and m > 2.


Holder and Lipschitz Regularity


Problem

Viscosity solutions of PIDEs are C 0,α/Lipschitz continuous in space (dep. on β)

||u||C 0,α ≤ C ||u||∞.

Main approaches for proving the Holder regularity of viscosity solutions oflocal/nonlocal equations

Harnack estimates, for uniformly elliptic equations: Guillen Schwab ’11,Silvestre Imbert ’14, Silvestre Cardaliaguet ’12, Silvestre Vicol ’12, Silvestre’06,’11, Caffarelli, Silvestre ’09, ’11 Caffarelli Cabre ’95Regularity and ABP estimates for a larger class of PIDEs left open!

direct viscosity methods such as Ishii-Lions’s ’90, for degenerate ellipticequations: Barles, Chasseigne and Imbert ’11, Cardaliaguet-Rainer ’10Lipschitz or further regularity, e.g. C 1,α left open!



Model equations for Holder and Lipschitz regularity

Advection fractional diffusion

ut + (−∆u)β/2 + b(x) · Du = f

Subcritical case β > 1: for b ∈ L∞ the solution is Lipschitz continuous.

Critical case β = 1: for b ∈ C τ , τ > 0 the solution is Cβ .

Supercritical case β < 1: for b ∈ C 1−β+τ , the solution is Cβ .



Model equations for Holder and Lipschitz regularity

Fractional difussion with superlinear gradient growth

ut − tr(A(x)D2u) + a2(x)(−∆)β/2u + b(x)|Du|k = f

with nondegenerate diffusion A(x) ≥ a1(x)I , with

a1(x) + a2(x) ≥ a0 > 0.

When β > 1: for b ∈ C 0,τ and k ≤ τ + β the solution is Lipschitz.

When β > 1: for b ∈ L∞ and k ≤ β the solution is Lipschitz continuous.

When β < 1: for b ∈ C 1−β+τ , and k ≤ β the solution is Cβ .




Mixed ellipticity

−a1(x1)∆x1u + a2(x2)(−∆x2u)β/2 = f (x1, x2)

with ai (x) ≥ a0 > 0.

Problem

Solutions are Lipschitz continuous in the x1-variable and Holder, resp. Lipschitzcontinuous in the x2 variable if β ≤ 1, resp. β > 1.



Partial Regularity

We give both Holder and Lipschitz regularity results of viscosity solutions for ageneral class of mixed integro-differential equations of the type

−a1(x1)∆x1u −a2(x2)Ix2 [x , u]− I[x , u]

+b1(x1)|Dx1u1|k1 +b2(x2)|Dx2u|k2 + |Du|n + cu = f (x).


Any periodic continuous viscosity solution u

(a) is Lipschitz in the x2 variable, if β > 1 and k2 ≤ β, k1 = 1, n ≥ 0;

(b) is C 0,α with α < β−k2

1−k2, if β ≤ 1 and k2 < β, k1 = 1, n ≥ 0.

(c) If b2 ∈ C 0,τ (Rd2 ), then we can deal wtih growth up to k2 ≤ β + τ .

The Lipschitz/Holder constant depends on ||u||∞, on the dimension d , theconstants associated to the Levy measures and on the functions a2, b2 and f .



Insight - Proof of the Regularity Results

Classical argument for Holder continuity: show that

maxx,y

(u(x)− u(y)− φ(|x − y |)) < 0.

where for Holder regularity the control function is given by

φ(|x − y |) = L|x − y |α

whereas for Lipschitz by





maxx,y

(u(x)− u(y)− φ(|x − y |)) < 0.


φ(|x − y |) = L|x − y |α


φ(|x − y |) = L|x − y |





maxx,y

(u(x)− u(y)− φ(|x − y |)) < 0.


φ(|x − y |) = L|x − y |α


φ(|x − y |) = L|x − y | − ρ|x − y |1+α.





maxx,y

(u(x)− u(y)− φ(|x − y |)) < 0.


φ(|x − y |) = L|x − y |α


φ(|x − y |) = L|x − y | − ρ|x − y |1+α.

For partial regularity, use classical regularity arguments in one set of variables,and uniqueness type arguments in the other variables:

ψε(x , y) = u(x1, x2)− u(y1, y2)− φ(x1 − y1)





maxx,y

(u(x)− u(y)− φ(|x − y |)) < 0.


φ(|x − y |) = L|x − y |α


φ(|x − y |) = L|x − y | − ρ|x − y |1+α.

For partial regularity, use classical regularity arguments in one set of variables,and uniqueness type arguments in the other variables:

ψε(x , y) = u(x1, x2)− u(y1, y2)− φ(x1 − y1)− |x2 − y2|2

ε2



Global regularity

A priori estimates. The regularity results can be extended to superlinear cases, bya gradient cut-off argument.

−a1(x1)∆x1u − a2(x2)Ix2 [x , u]− I[x , u]

+b1(x1)|Dx1u1| + b2(x2)|Dx2u| + |Du|n + cu = f (x).



(a) is Lipschitz continuous, if β > 1 and k2 = 1, k1 = 1, n ≥ 0;

(b) is C 0,α continuous with α < β2−k2

1−k2, if β ≤ 1 and k2 = 1, k1 = 1, n ≥ 0.



Global regularity

A priori estimates. The regularity results can be extended to superlinear cases, bya gradient cut-off argument.

−a1(x1)∆x1u − a2(x2)Ix2 [x , u]− I[x , u]

+b1(x1)|Dx1u1|k1 + b2(x2)|Dx2u|k2 + |Du|n + cu = f (x).



(a) is Lipschitz continuous, if β > 1 and k2 ≤ βi , k1 ≤ 2, n ≥ 0;

(b) is C 0,α continuous with α < β2−k2

1−k2, if β ≤ 1 and k2 < βi , k1 ≤ 2, n ≥ 0.



Extensions of the regularity results

the non-periodic setting

parabolic case

ut + F0(..., I[x , u]) + F1(..., Ix1 [x , u]) + F2(...,Jx2 [x , u]) = f (x)

fully nonlinear Bellman - Isaacs equations

supγ∈Γ

infδ∈∆

(F γ,δ0 (...,J γ,δ) + F γ,δ1 (...,J γ,δx1

) + F γ,δ2 (...,J γ,δx2)− f γ,δ(x)

)= 0

multiple nonlinearities.



Thank you for your attention!


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